Modeling Computation: INFO 2950 Prof. Carla Gomes gomes@cs.cornell.edu Module Modeling Computation: Language Recognition Rosen, Chapter 12.4

Models of computing DFA - regular languages Push down automata - Context-free Bounded Turing M’s - Context sensitive Turing machines - Phrase-structure

Foundations The theory of computation and the practical application it made possible — the computer — was developed by an Englishman called Alan Turing.

The Turing Machine (4) At any time, the machine is in one of a finite number of internal states. (5) The machine has instructions that determine what it does given its internal state and the symbol it encounters on the tape. It can  change its internal state;  change the symbol on the square;  move forward;  move backward;  halt (i.e. stop). Turing’s machine — which came to be called the Turing machine — was this: (1) A tape of infinite length (2) Finitely many squares of the tape have a single symbol from a finite language. (3) Someone (or something) that can read the squares and write in them.

1 1 1 1 Current state = 1 If current state = 1 and current symbol = 0 then new state = 10 new symbol = 1 move right 1 1 1 1

1 1 1 1 1 Current state = 10 If current state = 1 and current symbol = 0 then new state = 10 new symbol = 1 move right 1 1 1 1 1

1 1 1 1 1 Current state = 10 If current state = 1 and current symbol = 0 then new state = 10 new symbol = 1 move right 1 1 1 1 1

Functions It is essential to the idea of a Turing machine that it is not a physical machine, but an abstract one — a set of procedures. It makes no difference whether the machine is embodied by a person in a boxcar on a track, or a person with a paper and pencil, or a smart and well-trained flamingo.

First computers: custom computing machines Input tape (read only) Control Work tape (memory) 1950 -- Eniac: the control is hardwired manually for each problem. Output tape (write only) 1940: VON NEUMANN: DISTINCTION BETWEEN DATA AND INSTRUCTIONS

Can Machines Think? In “Computing machinery and intelligence,” written in 1950, Turing asks whether machines can think. He claims that this question is too vague, and proposes, instead, to replace it with a different one. That question is: Can machines pass the “imitation game” (now called the Turing test)? If they can, they are intelligent. Turing is thus the first to have offered a rigorous test for the determination of intelligence quite generally.

The Turing Test The game runs as follows. You sit at a computer terminal and have an electronic conversation. You don’t know who is on the other end; it could be a person or a computer responding as it has been programmed to do. If you can’t distinguish between a human being and a computer from your interactions, then the computer is intelligent. Note that this is meant to be a sufficient condition of intelligence only. There may be other ways to be intelligent.

The Church-Turning Thesis Turing, and a logician called Alonzo Church (1903-1995), independently developed the idea (not yet proven by widely accepted) that whatever can be computed by a mechanical procedure can be computed by a Turing machine. This is known as the Church-Turing thesis.

Turing Machines

The Language Hierarchy ? ? Context-Free Languages Regular Languages

Languages accepted by Turing Machines Context-Free Languages Regular Languages

A Turing Machine Tape ...... ...... Read-Write head Control Unit

The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right

...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

Example: Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left

Time 1 ...... ...... Time 2 ...... ...... 1. Reads 2. Writes 3. Moves Right

The Input String Input string Blank symbol ...... ...... head Head starts at the leftmost position of the input string

Input string Blank symbol ...... ...... head Remark: the input string is never empty

States & Transitions Write Read Move Left Move Right

Example: Time 1 ...... ...... current state

Time 1 ...... ...... Time 2 ...... ......

Halting The machine halts if there are no possible transitions to follow

Example: ...... ...... No possible transition HALT!!!

Not Allowed Final States Allowed Final states have no outgoing transitions In a final state the machine halts

Acceptance If machine halts in a final state Accept Input in a non-final state or If machine enters an infinite loop Reject Input

Turing Machine Example A Turing machine that accepts the language:

Time 0

Time 1

Time 2

Time 3

Time 4 Halt & Accept

Rejection Example Time 0

Time 1 No possible Transition Halt & Reject

Infinite Loop Example

Time 0

Time 1

Time 2

Time 2 Time 3 Infinite loop Time 4 Time 5

Because of the infinite loop: The final state cannot be reached The machine never halts The input is not accepted

Another Turing Machine Example Turing machine for the language

Time 0

Time 1

Time 2

Time 3

Time 4

Time 5

Time 6

Time 7

Time 8

Time 9

Time 10

Time 11

Time 12

Time 13 Halt & Accept

Languages accepted by Turing Machines Context-Free Languages Regular Languages

End of intro to Languages (sets of strings), Grammars, Automata, and Turing Machines (= computation).